Positive Eigenvalues of Circulant Hadamard Square Roots

Let $H$ be an entrywise positive real symmetric circulant matrix whose entrywise square is positive definite. We prove that $H$ has at least six positive eigenvalues if its order is at least $16$, and at least seven if its order is odd and at least $17$. The first threshold is sharp. The bounds are attained in order $16$ and in orders $17,19,21$, respectively. We also determine the minimum number of positive eigenvalues in every order from $5$ through $16$.